Zeta Functions and Green's Functions for Linear Partial Differential Operators of Elliptic Type with Constant Coefficients

1953 ◽  
Vol 57 (1) ◽  
pp. 32 ◽  
Author(s):  
S. Bochner
Keyword(s):  
Green's Functions ◽  
Differential Operators ◽  
Green’S Functions ◽  
Zeta Functions ◽  
Elliptic Type ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Constant Coefficients ◽  
Linear Partial Differential Operators

scholarly journals Generalized Vandermonde determinants for reversing Taylor's formula and application to hypoellipticity

2007 ◽  
Vol 38 (2) ◽  
pp. 183-189
Author(s):  
Giuseppe De Donno
Keyword(s):  
Differential Operators ◽  
Partial Differential ◽  
Taylor’S Formula ◽  
Several Variables ◽  
Partial Differential Operators ◽  
Constant Coefficients ◽  
Linear Partial Differential Operators ◽  
Necessary And Sufficient ◽  
Complex Coefficients ◽  
Taylor's Formula

The problem of the hypoellipticity of the linear partial differential operators with constant coefficients was completely solved by H"{o}r-man-der in [5]. He listed many equivalent algebraic conditions on the polynomial symbol of the operator, each necessary and sufficient for hypoellipticity. In this paper we employ two Mitchell's Theorems (1881) regarding a type of Generalized Vandermonde Determinants, for inverting Taylor's formula of polynomials in several variables with complex coefficients. We obtain then a more direct and easy proof of an equivalence for the mentioned H"{o}r-man-der's hypoellipticity conditions.

scholarly journals Some sufficient conditions for the comparability of two differential operators

Filomat ◽  
2002 ◽  
pp. 57-61 ◽  
Author(s):  
Raid Al-Momani ◽  
Qassem Al-Hassan ◽  
Ali Al-Jarrah ◽  
Ghanim Momani
Keyword(s):  
Partial Differential Equations ◽  
Differential Operators ◽  
Complete Solution ◽  
Sufficient Conditions ◽  
Order Relation ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
The Sixties ◽  
Constant Coefficients ◽  
Linear Partial Differential Operators

The comparison of differential operators is a problem of the theory of partial differential operators with constant coefficients. This problem up to now doesn't have a complete solution. It was formulated in the sixties by Lars Hormander in his monograph "The Analysis of Linear Partial Differential Operators". Many facts of the theory of partial differential equations can be formulated by using the concept of pre-order relation over the set of differential operators, however it is too complicated to check the comparability condition of two differential operators. In this paper we get some sufficient conditions for the comparability of two differential operators.

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